Solving Equations Step-by-Step A Justification Guide
In mathematics, solving equations is a fundamental skill. It involves a series of logical steps, each justified by specific properties and axioms. Understanding these justifications is as important as finding the solution itself. This article will thoroughly break down the solution process of the equation 2(3y + 4) - 3(y - 2) = 2y, justifying each step with the appropriate mathematical property. This step-by-step approach not only provides the solution but also enhances comprehension of the underlying principles, ensuring a solid foundation in algebraic problem-solving.
Original Equation
The original equation we aim to solve is:
This equation involves variables, constants, and arithmetic operations. Our goal is to isolate the variable y on one side of the equation to determine its value. Each step we take must be mathematically sound and justified by a property of equality or arithmetic.
Step 1 The Distributive Property
The first step in simplifying the equation is to apply the distributive property. This property states that for any real numbers a, b, and c, a(b + c) = ab + ac. In our equation, we have two terms that require distribution: 2(3y + 4) and -3(y - 2). Applying the distributive property to these terms allows us to eliminate the parentheses and combine like terms, which is crucial for simplifying the equation. This initial step sets the stage for subsequent operations, making it easier to isolate the variable and find the solution. The distributive property not only simplifies expressions but also helps in rearranging equations to a more manageable form, making it a cornerstone of algebraic manipulation. Understanding and correctly applying this property is essential for solving equations efficiently and accurately.
Applying the distributive property to the left side of the equation, we get:
2 * (3y) + 2 * (4) - 3 * (y) - 3 * (-2) = 2y$ $6y + 8 - 3y + 6 = 2y
Justification: Distributive Property of Multiplication over Addition and Subtraction
Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving multiplication and addition (or subtraction). It is expressed as a(b + c) = ab + ac, where a, b, and c can be any real numbers. This property essentially states that multiplying a number by a sum (or difference) is the same as multiplying the number by each term in the sum (or difference) individually and then adding (or subtracting) the results. In the context of equation solving, the distributive property is crucial for eliminating parentheses and combining like terms, making the equation easier to manipulate and solve. For instance, in the given equation, applying the distributive property to 2(3y + 4) transforms it into 6y + 8, effectively removing the parentheses and simplifying the expression. Similarly, applying it to -3(y - 2) yields -3y + 6, further simplifying the equation. The correct application of the distributive property is essential for maintaining the equality of the equation while simplifying it, and it forms the basis for many subsequent steps in the solution process. Mastering this property is therefore crucial for success in algebra and beyond.
Step 2 Combining Like Terms
After applying the distributive property, the next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. In our equation, 6y and -3y are like terms, and 8 and 6 are like terms. Combining like terms simplifies the equation by reducing the number of terms, which makes it easier to isolate the variable. This step is crucial for organizing the equation and making it more manageable. By grouping similar terms together, we can perform arithmetic operations to simplify the equation further. This process is not only essential for solving equations but also for simplifying algebraic expressions in general. Understanding how to identify and combine like terms is a fundamental skill in algebra, and it lays the groundwork for more advanced problem-solving techniques. The ability to combine like terms accurately ensures that the equation remains balanced and that the subsequent steps lead to the correct solution.
Combining the like terms 6y and -3y, and the constants 8 and 6, we get:
(6y - 3y) + (8 + 6) = 2y$ $3y + 14 = 2y
Justification: Combining Like Terms (Addition and Subtraction)
The Importance of Combining Like Terms
Combining like terms is a crucial step in solving algebraic equations because it simplifies the equation and makes it easier to manipulate. Like terms are those that contain the same variable raised to the same power; they can be added or subtracted together. For example, in the expression 6y - 3y + 8 + 6, the terms 6y and -3y are like terms, as are the constants 8 and 6. Combining like terms involves performing the indicated arithmetic operations on the coefficients of the like terms while keeping the variable part the same. In this case, 6y - 3y simplifies to 3y, and 8 + 6 simplifies to 14. The simplified expression becomes 3y + 14. This process reduces the number of terms in the equation, making it less complex and easier to solve. Combining like terms is based on the distributive property in reverse; for instance, 6y - 3y can be seen as (6 - 3)y, which simplifies to 3y. This simplification is essential because it groups similar terms together, allowing us to isolate the variable more effectively and ultimately find the solution to the equation. Without combining like terms, the equation would be more cumbersome to work with, increasing the chances of making errors in subsequent steps.
Step 3 Isolating the Variable
Now, our goal is to isolate the variable y on one side of the equation. To do this, we need to eliminate the term 2y from the right side of the equation. The Subtraction Property of Equality allows us to subtract the same value from both sides of an equation without changing its balance. By subtracting 2y from both sides, we can move all the y terms to one side of the equation, making it closer to the isolated form. This step is critical in simplifying the equation and bringing us closer to the solution. Isolating the variable is a fundamental technique in algebra, and it relies on the principle that performing the same operation on both sides of an equation maintains equality. Understanding and applying this property correctly is essential for solving various types of equations. The subtraction property of equality is one of several properties that allow us to manipulate equations while preserving their validity, and it is a cornerstone of algebraic problem-solving.
To isolate the variable, subtract 2y from both sides:
3y + 14 - 2y = 2y - 2y$ $y + 14 = 0
Justification: Subtraction Property of Equality
The Subtraction Property of Equality
The Subtraction Property of Equality is a fundamental principle in algebra that states if you subtract the same quantity from both sides of an equation, the equation remains balanced. This property is crucial for isolating variables and solving equations. Mathematically, it can be expressed as follows: if a = b, then a - c = b - c for any real numbers a, b, and c. In the context of solving equations, this property allows us to eliminate terms from one side of the equation by performing subtraction, effectively moving terms from one side to the other. For example, if we have an equation like 3y + 14 = 2y, we can subtract 2y from both sides to simplify the equation and group the y terms on one side. This results in 3y + 14 - 2y = 2y - 2y, which simplifies to y + 14 = 0. The subtraction property of equality ensures that the equation remains valid after this operation, and it is a key tool for simplifying equations and isolating variables. Without this property, it would be much more challenging to manipulate equations and solve for unknowns. Understanding and applying the subtraction property of equality correctly is essential for mastering algebraic problem-solving.
Step 4 Isolating the Variable (Continued)
To completely isolate the variable y, we need to eliminate the constant term 14 from the left side of the equation. Again, we use the Subtraction Property of Equality. By subtracting 14 from both sides of the equation, we can isolate y on the left side, giving us the solution. This final step demonstrates the importance of performing inverse operations to solve for a variable. The subtraction property of equality is a powerful tool that allows us to manipulate equations while maintaining their balance. It is a cornerstone of algebraic problem-solving and is used extensively in various mathematical contexts. Understanding and applying this property correctly is crucial for solving equations efficiently and accurately. Isolating the variable is the ultimate goal in solving equations, and this step brings us to the final answer by removing any remaining terms from the variable's side.
Subtract 14 from both sides:
y + 14 - 14 = 0 - 14$ $y = -14
Justification: Subtraction Property of Equality
Final Thoughts on Isolating the Variable
Isolating the variable is the ultimate goal in solving algebraic equations, and it involves a series of strategic steps to get the variable alone on one side of the equation. This process typically involves using inverse operations to undo any operations that are being applied to the variable. For instance, if the variable is being added to a constant, we subtract that constant from both sides of the equation. Similarly, if the variable is being multiplied by a coefficient, we divide both sides of the equation by that coefficient. The key principle underlying these steps is the properties of equality, which ensure that the equation remains balanced throughout the solution process. The Addition, Subtraction, Multiplication, and Division Properties of Equality allow us to perform the same operation on both sides of the equation without changing its solution. Isolating the variable often requires combining several steps, such as distributing, combining like terms, and applying inverse operations. Each step must be justified by a property of equality or a mathematical rule. Once the variable is isolated, the value on the other side of the equation represents the solution. This process is fundamental to solving a wide range of algebraic problems, and mastering it is essential for success in mathematics.
Solution
Therefore, the solution to the equation is:
This solution can be verified by substituting -14 back into the original equation to ensure that both sides are equal. Verification is a crucial step in problem-solving as it confirms the accuracy of the solution and helps identify any potential errors in the process. By substituting the solution back into the original equation, we can gain confidence in our answer and ensure that we have followed the correct steps. This practice is not only beneficial for individual problem-solving but also for developing a deeper understanding of the equation-solving process and the properties that govern it. Verification reinforces the connection between the solution and the original problem, highlighting the logical consistency of the solution.
Verifying the Solution
Verifying the solution to an equation is a crucial step in the problem-solving process, as it confirms the accuracy of the answer. This process involves substituting the solution back into the original equation to ensure that both sides of the equation are equal. If the solution is correct, the substitution will result in a true statement, demonstrating that the solution satisfies the equation. For example, if we solve an equation and find that y = -14, we would substitute -14 for y in the original equation. If the left side of the equation equals the right side after this substitution, then -14 is indeed the correct solution. Verification is particularly important in algebra, where equations can become complex, and errors can easily occur. It not only provides assurance that the solution is correct but also helps identify any mistakes made during the solution process. By verifying solutions, students develop a deeper understanding of the relationship between equations and their solutions, and they enhance their problem-solving skills. This practice reinforces the idea that the solution should make the equation true, and it promotes careful and accurate work in mathematics. Moreover, verification can help in identifying conceptual errors or misunderstandings, leading to better learning and mastery of algebraic concepts.
Conclusion
In conclusion, solving the equation 2(3y + 4) - 3(y - 2) = 2y involves several steps, each justified by mathematical properties. We started by applying the distributive property, then combined like terms, and finally used the Subtraction Property of Equality to isolate the variable and find the solution y = -14. Understanding these steps and their justifications is crucial for mastering algebraic problem-solving. The ability to solve equations accurately and efficiently is a fundamental skill in mathematics, and it forms the basis for more advanced concepts and applications. By breaking down the solution process into smaller, justified steps, we gain a deeper understanding of the underlying principles and develop confidence in our problem-solving abilities. This step-by-step approach not only leads to the correct answer but also enhances our mathematical reasoning and critical thinking skills. Mastering these techniques is essential for success in algebra and beyond.
Final Thoughts on Equation Solving
Solving equations is a cornerstone of mathematics and a critical skill for anyone studying algebra and beyond. The process involves a systematic approach, where each step is carefully considered and justified by mathematical properties. Understanding these properties is as important as finding the solution itself, as it provides a solid foundation for problem-solving and enhances mathematical reasoning. Equations are mathematical statements that assert the equality of two expressions, and solving them means finding the value(s) of the variable(s) that make the statement true. The basic properties of equality, such as the Addition, Subtraction, Multiplication, and Division Properties, are the fundamental tools for manipulating equations. These properties allow us to perform the same operation on both sides of the equation without changing its solution. Other important techniques include the distributive property, which allows us to simplify expressions involving parentheses, and combining like terms, which simplifies equations by grouping similar terms together. The process of solving equations often requires a combination of these techniques, and it is essential to follow a logical and organized approach. Verification of the solution is a crucial final step, as it ensures the accuracy of the answer. By mastering equation-solving skills, students develop the ability to think logically, solve problems systematically, and apply mathematical concepts in various contexts. This skill is not only essential for academic success but also for real-world problem-solving.
